More Effective Criteria for Oscillation of Second-Order Differential Equations with Neutral Arguments

Abstract

The motivation for this paper is to create new criteria for oscillation of solutions of second-order nonlinear neutral differential equations. In more than one respect, our results improve several related ones in the literature. As proof of the effectiveness of the new criteria, we offer more than one practical example.

1. Introduction

In recent decades, an increasing interest in establishing sufficient criteria for oscillatory and non-oscillatory properties of different classes of differential equations has been observed; see, for instance, the monographs [1,2,3] and the references cited therein. Many authors were concerned with the oscillation and nonoscillation of delay differential equations of second-order [4,5,6,7,8,9,10,11,12,13,14,15,16,17] and higher-order [18,19,20,21,22,23]. The growing interest of delay differential equation is due to the many applications of these equations in different fields of science, see [24,25].

In this work, by using different techniques, we create new criteria for oscillation of 2nd-order neutral differential equation

$${\left(r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }\right)}^{\prime }+q\left(\zeta \right)x^{\beta }\left(\sigma \left(\zeta \right)\right)=0,$$

(1)

where $\zeta \ge {\zeta }_0$ and $z\left(\zeta \right):=x\left(\zeta \right)+p\left(\zeta \right)x\left(\tau \left(\zeta \right)\right)$. Moreover, we assume that:

[N₁]

$\alpha ,\beta \in {\mathbb{Q}}_{odd}^{+},$ where ${\mathbb{Q}}_{odd}^{+}:=\left\{a/b:a,b\in {\mathbb{Z}}^{+}\mathrm{are}\mathrm{odd}\right\}$;

[N₂]

$r\in C\left([{\zeta }_0,\infty )\right),$$r\left(\zeta \right)>0$ and

$${\theta }_s\left(\zeta \right):={\int }_s^{\zeta }{r}^{-1/\alpha }\left(s\right)\mathrm{d}s;$$

[N₃]

$p,q\in C\left([{\zeta }_0,\infty )\right),$$q\left(\zeta \right)\ge 0,$$0\le p\left(\zeta \right)\le p_0<\infty$ and $q\left(\zeta \right)$ is not identically zero for large $\zeta$;

[N₄]

$\tau ,\sigma \in C\left([{\zeta }_0,\infty )\right),$$\tau \left(\zeta \right)\le \zeta ,$$\sigma \left(\zeta \right)<\zeta ,$ and ${lim}_{\zeta \to \infty }\tau \left(\zeta \right)={lim}_{\zeta \to \infty }\sigma \left(\zeta \right)=\infty .$

If x$\in C^1[{\zeta }_x,\infty ),{\zeta }_x\ge {\zeta }_0,$$r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }\in C^1[{\zeta }_x,\infty )$ for all ${\zeta }_x\ge {\zeta }_0$, and x satisfies (1) on $[{\zeta }_x,\infty )$, then x is called a solution of (1). Our attention will be solely to the solutions which satisfy $sup\{\left|x\left(\zeta \right)\right|:\zeta \ge \zeta \}>0,$ for all $\zeta \ge {\zeta }_x$. If there exists a ${\zeta }_1\ge {\zeta }_0$ such that either $x\left(\zeta \right)>0$ or $x\left(\zeta \right)<0$ for all $\zeta \ge {\zeta }_1$, then x is said to be a non-oscillatory solution; otherwise, it is said to be an oscillatory solution.

First, we will shed light on the studies of canonical equations that require ${\theta }_{{\zeta }_0}\left(\infty \right)=\infty$. By employing the Riccati transformation, Sun and Meng [16] studied the oscillation of delay Equation (1), with $p\left(\zeta \right)=0$ and $\alpha =\beta$. They proved that (1) is oscillatory if

$${\int }_{{\zeta }_1}^{\infty }\left({\theta }_{{\zeta }_1}^{\alpha }\left(\sigma \left(s\right)\right)q\left(s\right)-\frac{{\alpha }^{\alpha +1}{\sigma }^{\prime }\left(s\right)}{{\left(\alpha +1\right)}^{\alpha +1}{\theta }_{{\zeta }_1}\left(\sigma \left(s\right)\right)r^{1/\alpha }\left(\sigma \left(s\right)\right)}\right)\mathrm{d}s=\infty .$$

Likewise, Xu and Meng [17] extended the results of [16] to the neutral case, and proved that if

$${\int }_{{\zeta }_1}^{\infty }\left({\theta }_{{\zeta }_1}^{\alpha }\left(\sigma \left(s\right)\right)G\left(s\right)-\frac{{\alpha }^{\alpha +1}{\sigma }^{\prime }\left(s\right)}{{\left(\alpha +1\right)}^{\alpha +1}{\theta }_{{\zeta }_1}\left(\sigma \left(s\right)\right)r^{1/\alpha }\left(\sigma \left(s\right)\right)}\right)\mathrm{d}s=\infty ,\alpha =\beta ,$$

then (1) is oscillatory, where $G\left(\zeta \right):=q\left(\zeta \right){\left(1-p\left(\sigma \left(\zeta \right)\right)\right)}^{\alpha }$.

By a different approach, by using comparison theorems that compare the second-order equation with first-order equations, Baculikova and Dzurina [5] proved that (1) is oscillatory if $\alpha \ge \beta ,$ $\sigma \left(\zeta \right)\le \tau \left(\zeta \right)$ and

$$\underset{\zeta \to \infty }{liminf}{\int }_{{\tau }^{-1}\left(\sigma \left(\zeta \right)\right)}^{\zeta }\widehat{G}\left(s\right){\left({\int }_{{\zeta }_1}^{\sigma \left(\zeta \right)}{r}^{-1/\alpha }\left(u\right)\mathrm{d}u\right)}^{\beta }\mathrm{d}s>{\left(1+\frac{p_0^{\beta }}{{\tau }_0}\right)}^{\beta /\alpha }\frac{1}{\kappa \mathrm{e}},$$

(2)

where

$$\widehat{G}\left(\zeta \right):=min\left\{q\left(\zeta \right),q\left(\tau \left(\zeta \right)\right)\right\}$$

(3)

and

$$\kappa :=\left\{\begin{array}{cc}1\hfill & \mathrm{if}0<\beta \le 1;\\ 2^{1-\beta }\hfill & \mathrm{if}\beta >1.\end{array}\right.$$

(4)

Very recently, complementing the approach taken in [7], Grace et al. [11] improved the results in [5,13,17]. They established the following criteria for oscillation of (1) with $\alpha =\beta$:

(a)

By using comparison theory:

$$\underset{\zeta \to \infty }{liminf}{\int }_{\sigma \left(\zeta \right)}^{\zeta }G\left(s\right){\left({\theta }_{{\zeta }_1}^{*}\left(\sigma \left(s\right)\right)\right)}^{\alpha }\mathrm{d}s>\frac{1}{\mathrm{e}},$$

where

$${\theta }_{{\zeta }_1}^{*}\left(\zeta \right):={\theta }_{{\zeta }_1}\left(\zeta \right)+\frac{1}{\alpha }{\int }_{{\zeta }_1}^{\zeta }{\theta }_{{\zeta }_1}\left(u\right){\theta }_{{\zeta }_1}^{\alpha }\left(u\right)\mathrm{d}u;$$

(b)

By employing the Riccati transformation

$$\underset{\zeta \to \infty }{limsup}{\int }_{{\zeta }_1}^{\infty }\left(\varphi \left(s\right)exp\left(-{\int }_{\sigma \left(s\right)}^s\frac{\mathrm{d}u}{r^{1/\alpha }\left(u\right){\theta }_{{\zeta }_1}^{*}\left(u\right)}\right)-\frac{r\left(s\right){\left({\varphi }_{+}^{\prime }\left(s\right)\right)}^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}{\varphi }^{\alpha }\left(s\right)}\right)\mathrm{d}s=\infty ,$$

(5)

where $\varphi \in C\left(\left[{\zeta }_0,\infty \right),\left(0,\infty \right)\right)$ and ${\varphi }_{+}^{\prime }\left(\zeta \right):=max\left\{{\varphi }^{\prime }\left(\zeta \right),0\right\}.$

Moreover, Moaaz [14] extended the results of [11] to (1) when $\alpha >\beta$ and $\alpha <\beta$.

On the other hand, there are many studies to improve the criteria for oscillation of solutions of non-canonical equations ${\theta }_{{\zeta }_0}\left(\infty \right)<\infty$, some of which we will refer to below.

Recently, Agarwal et al. [4] established conditions for oscillation of (1) which are an improvement for its predecessors. They proved that (1), with $\beta =\alpha$, is oscillatory if

$$\underset{\zeta \to \infty }{limsup}{\int }_{{\zeta }_0}^{\zeta }\left(\rho \left(s\right)q\left(s\right){\left(1-p\left(\sigma \left(s\right)\right)\right)}^{\alpha }-\frac{{\left({\left({\rho }^{\prime }\left(s\right)\right)}_{+}\right)}^{\alpha +1}r\left(\tau \left(s\right)\right)}{{\left(\alpha +1\right)}^{\alpha +1}{\rho }^{\alpha }\left(s\right){\left({\tau }^{\prime }\left(s\right)\right)}^{\alpha }}\right)\mathrm{d}s=\infty$$

(6)

and

$$\underset{\zeta \to \infty }{limsup}{\int }_{{\zeta }_0}^{\zeta }\left(\psi \left(s\right)-\frac{\delta \left(s\right)r\left(s\right){\left({\left(\phi \left(s\right)\right)}_{+}\right)}^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}}\right)\mathrm{d}s=\infty$$

(7)

hold, where

$$\psi \left(\zeta \right)=\delta \left(\zeta \right)\left(q\left(\zeta \right){\left(1-p\left(\sigma \left(\zeta \right)\right)\frac{\pi \left(\tau \left(\sigma \left(\zeta \right)\right)\right)}{\pi \left(\sigma \left(\zeta \right)\right)}\right)}^{\alpha }+\frac{1-\alpha }{r^{1/\alpha }\left(\zeta \right){\pi }^{\alpha +1}\left(\zeta \right)}\right),$$

and $\phi \left(\zeta \right)=\left({\delta }^{\prime }\left(\zeta \right)/\delta \left(\zeta \right)\right)+\left(1+\alpha \right)/\left(r^{1/\alpha }\left(\zeta \right)\pi \left(\zeta \right)\right)$ and ${\left(\phi \left(\zeta \right)\right)}_{+}=max\{\phi \left(\zeta \right),0\}.$

Bohner et al. in [7], Theorem 2.4 simplified the criteria for oscillatory by setting one sufficient condition

$$\underset{\zeta \to \infty }{liminf}{\int }_{\sigma \left(\zeta \right)}^{\zeta }{\left(\frac{1}{r\left(\zeta \right)}{\int }_{{\zeta }_1}^{\zeta }q\left(\zeta \right){\left(1-p\left(\sigma \left(\zeta \right)\right)\frac{\pi \left(\tau \left(\sigma \left(\zeta \right)\right)\right)}{\pi \left(\sigma \left(\zeta \right)\right)}\right)}^{\alpha }\right)}^{1/\alpha }\mathrm{d}s>\frac{1}{\mathrm{e}}.$$

The main purpose of this work is to improve conditions (2) and (5) by establishing a new criterion for oscillation of (1), which also takes into account the influence of delay argument $\tau \left(\zeta \right)$. Our approach is essentially based on presenting sharper criteria for oscillation solutions of (1) than criteria in [5,11]. Moreover, for non-canonical case, we improve and complete some results in [4,7]. As proof of the effectiveness of the new criteria, we offer more than one practical example.

2. Oscillation Theorems in Canonical Case

In this section, we establish new criteria for oscillation of solution of (1) in canonical case ${\theta }_{{\zeta }_0}\left(\infty \right)=\infty$. For convenience, we denote that: $Q\left(\zeta \right):=q\left(\zeta \right){\left(1-p\left(\sigma \left(\zeta \right)\right)\right)}^{\beta },$

$$\begin{array}{ccc}\hfill {\tilde{\theta }}_{{\zeta }_0}\left(\zeta \right)& :& ={\theta }_{{\zeta }_0}\left(\zeta \right)+\frac{1}{\alpha }{\int }_{{\zeta }_0}^{\zeta }{\theta }_{{\zeta }_0}\left(u\right){\theta }_{{\zeta }_0}^{\alpha }\left(\sigma \left(\zeta \right)\right)\eta \left(\sigma \left(u\right)\right)Q\left(u\right)\mathrm{d}u,\hfill \\ \hfill {\widehat{\theta }}_{{\zeta }_0}\left(\zeta \right)& :& =exp\left(-\alpha {\int }_{\sigma \left(\zeta \right)}^{\zeta }\frac{1}{{\tilde{\theta }}_{{\zeta }_0}\left(s\right)r^{1/\alpha }\left(s\right)}\mathrm{d}s\right),\hfill \\ \hfill {\psi }_k\left(\zeta \right)& :& ={\int }_{\zeta }^{\infty }{\widehat{\theta }}^k\left(u\right)\eta \left(\sigma \left(u\right)\right)Q\left(u\right)\mathrm{d}u,k=0,1,\hfill \end{array}$$

and

$$\eta \left(\zeta \right):=\left\{\begin{array}{cc}{c}_1^{\beta -\alpha }\hfill & \mathrm{if}\alpha \le \beta ;\\ c_2{\theta }_{{\zeta }_2}^{\beta -\alpha }\left(\zeta \right)\hfill & \mathrm{if}\alpha >\beta ,\end{array}\right.$$

where $c_1$ and $c_2$ are positive constants.

Lemma 1.

Assume that x is an eventually positive solution of (1). Then,

$$z\left(\zeta \right)>0,z^{\prime }\left(\zeta \right)>0\mathit{and}\left(r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }\right)\le 0,$$

(8)

for $\zeta \ge {\zeta }_1$, where ${\zeta }_1$ is sufficiently large. Moreover, $z^{\beta -\alpha }\left(\zeta \right)\ge \eta \left(\zeta \right)$, eventually.

Proof. 

First, we postulate that x is a positive solution of (1). From [5], Lemma 3, we have that (8) holds.

Next, let $\alpha \le \beta$. From the monotonicity of z, we get that $z\left(\zeta \right)\ge z\left({\zeta }_2\right):=c_1>0$ for $\zeta \ge {\zeta }_2,$ where ${\zeta }_2$ is sufficiently large.

In the case where $\alpha >\beta$, since $r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }$ is non-increasing, we have that

$$z\left(\zeta \right)\le z\left({\zeta }_2\right)+r^{1/\alpha }\left({\zeta }_2\right)z^{\prime }\left({\zeta }_2\right){\theta }_{{\zeta }_2}\left(\zeta \right),$$

for $\zeta \ge {\zeta }_2$. In view of the fact that ${\theta }_{{\zeta }_2}\left(\infty \right)=\infty$, there exists a $N>0$ and ${\zeta }_N>{\zeta }_2$ such that ${\theta }_{{\zeta }_2}\left(\zeta \right)>N$ for all $\zeta \ge {\zeta }_N$. Thus, $z\left(\zeta \right)\le c_2{\theta }_{{\zeta }_2}\left(\zeta \right)$ where $c_3:=\left(z\left({\zeta }_2\right)/N+r^{1/\alpha }\left({\zeta }_2\right)z^{\prime }\left({\zeta }_2\right)\right)$. Consequently, $z^{\beta -\alpha }\left(\zeta \right)\ge \eta \left(\zeta \right)$. The proof is complete. □

Theorem 1.

Assume that

$$\sigma \left(\zeta \right)\le \tau \left(\zeta \right),{\tau }^{\prime }\left(\zeta \right)\ge {\tau }_0>0\mathrm{and}\tau \circ \sigma =\sigma \circ \tau .$$

If

$$\underset{\zeta \to \infty }{liminf}{\int }_{{\tau }^{-1}\left(\sigma \left(\zeta \right)\right)}^{\zeta }\widehat{G}\left(s\right){\tilde{\theta }}_{{\zeta }_1}^{\beta }\left(\sigma \left(\zeta \right)\right)\mathrm{d}s>{\left(1+\frac{p_0^{\beta }}{{\tau }_0}\right)}^{\beta /\alpha }\frac{1}{\kappa \mathrm{e}},$$

(9)

where $\widehat{G}\left(s\right)$ and κ defined as in (3) and (4), respectively, then all solutions of (1) are oscillatory.

Proof. 

Suppose the contrary; that (1) has an eventually non-oscillatory solution. Without loss of generality, we assume that $x\left(\zeta \right)>0,$$x\left(\tau \left(\zeta \right)\right)>0$ and $x\left(\sigma \left(\zeta \right)\right)>0$ for $\zeta \ge {\zeta }_1,$ where ${\zeta }_1$ is sufficiently large. By Lemma 1, we have that (8) holds. Taking (8) into account, we obtain $x\left(\zeta \right)\ge z\left(\zeta \right)\left(1-p\left(\sigma \left(\zeta \right)\right)\right)$, which with (1) gives

$${\left(r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }\right)}^{\prime }\le -Q\left(\zeta \right)z^{\beta }\left(\sigma \left(\zeta \right)\right).$$

(10)

Using the chain rule and simple computation, we see that

$$\alpha {\left(r^{1/\alpha }{z}^{\prime }\right)}^{\alpha -1}\frac{\mathrm{d}}{\mathrm{d}\zeta }\left(z-{\theta }_{{\zeta }_1}{r}^{1/\alpha }{z}^{\prime }\right)=-\alpha {\left(r^{1/\alpha }{z}^{\prime }\right)}^{\alpha -1}{\theta }_{{\zeta }_1}{\left(r^{1/\alpha }{z}^{\prime }\right)}^{\prime }=-{\theta }_{{\zeta }_1}{\left(r{\left(z^{\prime }\right)}^{\alpha }\right)}^{\prime }.$$

(11)

From (10) and (11), we obtain

$$\begin{array}{ccc}\frac{\mathrm{d}}{\mathrm{d}\zeta }\left(z\left(\zeta \right)-{\theta }_{{\zeta }_1}\left(\zeta \right)r^{1/\alpha }\left(\zeta \right)z^{\prime }\left(\zeta \right)\right)& \ge & \frac{1}{\alpha }{\left(r^{1/\alpha }\left(\zeta \right)z^{\prime }\left(\zeta \right)\right)}^{1-\alpha }{\theta }_{{\zeta }_1}\left(\zeta \right)Q\left(\zeta \right)z^{\beta }\left(\sigma \left(\zeta \right)\right)\hfill \\ & \ge & \frac{1}{\alpha }{\left(r^{1/\alpha }\left(\zeta \right)z^{\prime }\left(\zeta \right)\right)}^{1-\alpha }{\theta }_{{\zeta }_1}\left(\zeta \right)Q\left(\zeta \right)\eta \left(\sigma \left(\zeta \right)\right)z^{\alpha }\left(\sigma \left(\zeta \right)\right).\hfill \end{array}$$

Integrating this inequality from ${\zeta }_1$ to $\zeta$, we arrive at

$$z\left(\zeta \right)\ge {\theta }_{{\zeta }_1}\left(\zeta \right)r^{1/\alpha }\left(\zeta \right)z^{\prime }\left(\zeta \right)+\frac{1}{\alpha }{\int }_{{\zeta }_1}^{\zeta }{\left(r^{1/\alpha }\left(u\right)z^{\prime }\left(u\right)\right)}^{1-\alpha }{\theta }_{{\zeta }_1}\left(u\right)Q\left(u\right)\eta \left(\sigma \left(u\right)\right)z^{\alpha }\left(\sigma \left(u\right)\right)\mathrm{d}u.$$

(12)

From the monotonicity of $r^{1/\alpha }\left(\zeta \right)z^{\prime }\left(\zeta \right)$, we have

$$z\left(\sigma \left(\zeta \right)\right)\ge {\theta }_{{\zeta }_1}\left(\sigma \left(\zeta \right)\right)r^{1/\alpha }\left(\sigma \left(\zeta \right)\right)z^{\prime }\left(\sigma \left(\zeta \right)\right)\ge {\theta }_{{\zeta }_1}\left(\sigma \left(\zeta \right)\right)r^{1/\alpha }\left(\zeta \right)z^{\prime }\left(\zeta \right).$$

Thus, (12) becomes

$$z\left(\zeta \right)\ge {\tilde{\theta }}_{{\zeta }_1}\left(\zeta \right)r^{1/\alpha }\left(\zeta \right)z^{\prime }\left(\zeta \right).$$

(13)

Proceeding as in the proof of Theorem 1 in [5] and using (13) instead of ((2.10) in [5]), we arrive at

$${\left(\omega \left(\zeta \right)+\frac{p_0^{\beta }}{{\tau }_0}\omega \left(\tau \left(\zeta \right)\right)\right)}^{\prime }+\kappa \widehat{G}\left(\zeta \right){\tilde{\theta }}_{{\zeta }_1}^{\beta }\left(\sigma \left(\zeta \right)\right){\omega }^{\beta /\alpha }\left(\sigma \left(\zeta \right)\right)\le 0,$$

where $\omega :=r^{1/\alpha }{z}^{\prime }$. Next, set $\psi \left(\zeta \right):=\omega \left(\zeta \right)+\left(p_0^{\beta }/{\tau }_0\right)\omega \left(\tau \left(\zeta \right)\right)>0.$ Since $\tau \left(\zeta \right)\le \zeta$ and ${\omega }^{\prime }\left(t\right)\le 0$, we obtain

$$\begin{array}{ccc}\psi \left(\zeta \right)& \le & \omega \left(\tau \left(\zeta \right)\right)+\left(p_0^{\beta }/{\tau }_0\right)\omega \left(\tau \left(\zeta \right)\right)\hfill \\ & \le & \omega \left(\tau \left(\zeta \right)\right)\left(1+\frac{p_0^{\beta }}{{\tau }_0}\right).\hfill \end{array}$$

Thus, $\psi$ is a positive solution of

$${\psi }^{\prime }\left(\zeta \right)+\kappa {\left(\frac{{\tau }_0}{{\tau }_0+p_0^{\beta }}\right)}^{\beta /\alpha }\widehat{G}\left(\zeta \right){\tilde{\theta }}_{{\zeta }_1}^{\beta }\left(\sigma \left(\zeta \right)\right){\psi }^{\beta /\alpha }\left({\tau }^{-1}\left(\sigma \left(\zeta \right)\right)\right)\le 0.$$

From Theorem 1 in [26], the equation

$${\psi }^{\prime }\left(\zeta \right)+\kappa {\left(\frac{{\tau }_0}{{\tau }_0+p_0^{\beta }}\right)}^{\beta /\alpha }\widehat{G}\left(\zeta \right){\tilde{\theta }}_{{\zeta }_1}^{\beta }\left(\sigma \left(\zeta \right)\right){\psi }^{\beta /\alpha }\left({\tau }^{-1}\left(\sigma \left(\zeta \right)\right)\right)=0,$$

(14)

also has a positive solution. It is well-known (see, e.g., [27], Theorem 2) that condition (9) implies oscillation of (14). This contradiction completes the proof. □

Theorem 2.

If

$$\underset{\zeta \to \infty }{liminf}\frac{\alpha }{{\psi }_1\left(\zeta \right)}{\int }_{\zeta }^{\infty }{r}^{-1/\alpha }\left(u\right){\psi }_1^{\left(\alpha +1\right)/\alpha }\left(u\right)\mathrm{d}u>\frac{\alpha }{{\left(\alpha +1\right)}^{\left(\alpha +1\right)/\alpha }},$$

(15)

then all solutions of (1) are oscillatory.

Proof. 

Proceeding as in the proof of Theorem 1, we arrive at (13). From (13), we see that

$$\frac{z\left(\sigma \left(\zeta \right)\right)}{z\left(\zeta \right)}\ge exp\left(-{\int }_{\sigma \left(\zeta \right)}^{\zeta }\frac{1}{{\tilde{\theta }}_{{\zeta }_1}\left(s\right)r^{1/\alpha }\left(s\right)}\mathrm{d}s\right).$$

(16)

Define the function

$$w\left(\zeta \right):=\frac{r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }}{z^{\alpha }\left(\zeta \right)}.$$

(17)

Then $\omega \left(\zeta \right)>0$ for $\zeta \ge {\zeta }_1$. From (1) and (17), we get

$$w^{\prime }\left(\zeta \right)\le -Q\left(\zeta \right)\frac{z^{\alpha }\left(\sigma \left(\zeta \right)\right)}{z^{\alpha }\left(\zeta \right)}{z}^{\beta -\alpha }\left(\sigma \left(\zeta \right)\right)-\frac{\alpha }{r^{1/\alpha }\left(\zeta \right)}{w}^{\left(\alpha +1\right)/\alpha }\left(\zeta \right).$$

Using Lemma 1 and (16), we obtain

$$w^{\prime }\left(\zeta \right)\le -\widehat{\theta }\left(\zeta \right)\eta \left(\sigma \left(\zeta \right)\right)Q\left(\zeta \right)-\frac{\alpha }{r^{1/\alpha }\left(\zeta \right)}{w}^{\left(\alpha +1\right)/\alpha }\left(\zeta \right)<0.$$

(18)

By integrating (18) from $\zeta$ to $s,$ we conclude that

$${\int }_{\zeta }^s\widehat{\theta }\left(u\right)\eta \left(\sigma \left(u\right)\right)Q\left(u\right)\mathrm{d}u+\alpha {\int }_{\zeta }^s{r}^{-1/\alpha }\left(u\right)w^{\left(\alpha +1\right)/\alpha }\left(u\right)\mathrm{d}u\le w\left(\zeta \right)-w\left(s\right).$$

Since w is positive decreasing function, we see that

$${\psi }_1\left(\zeta \right)+\alpha {\int }_{\zeta }^{\infty }{r}^{-1/\alpha }\left(u\right)w^{\left(\alpha +1\right)/\alpha }\left(u\right)\mathrm{d}u\le w\left(\zeta \right).$$

Hence,

$$1+\frac{\alpha }{{\psi }_1\left(\zeta \right)}{\int }_{\zeta }^{\infty }{r}^{-1/\alpha }\left(u\right){\psi }^{\left(\alpha +1\right)/\alpha }\left(u\right){\left(\frac{w\left(u\right)}{\psi \left(u\right)}\right)}^{\left(\alpha +1\right)/\alpha }\mathrm{d}u\le \frac{w\left(\zeta \right)}{{\psi }_1\left(\zeta \right)}.$$

(19)

Set

$$\delta :=\underset{\zeta \ge {\zeta }_1}{inf}\frac{w\left(\zeta \right)}{{\psi }_1\left(\zeta \right)}.$$

From (19), $\delta \ge 1$. Taking (15) and (19) into account, we find

$$1+\alpha {\left(\frac{\delta }{\alpha +1}\right)}^{1+1/\alpha }\le \delta$$

or

$${\left(\frac{\delta }{\alpha +1}\right)}^{\alpha +1}\le {\left(\frac{\delta -1}{\alpha }\right)}^{\alpha },$$

which is not possible with the permissible value $\alpha >0$ and $\delta \ge 1$. This contradiction completes the proof. □

In the following, we give an example to illustrate our main results.

Example 1.

Consider the differential equation

$${\left({\left({\left(x\left(\zeta \right)+p_{0}x\left(\mu \zeta \right)\right)}^{\prime }\right)}^{\alpha }\right)}^{\prime }+\frac{q_0}{{\zeta }^{\alpha +1}}{x}^{\alpha }\left(\lambda \zeta \right)=0,$$

(20)

where $q_0>0$ and $\mu ,\lambda \in \left(0,1\right)$. We note that ${\theta }_{{\zeta }_0}\left(\zeta \right)=\zeta ,$

$$Q\left(\zeta \right)=\frac{q_0}{{\zeta }^{\alpha +1}}{\left(1-p_0\right)}^{\alpha },\widehat{G}\left(\zeta \right):=\frac{q_0}{{\zeta }^{\alpha +1}},{\tilde{\theta }}_{{\zeta }_0}\left(\zeta \right)=A\zeta ,\widehat{\theta }\left(\zeta \right)={\lambda }^{\alpha /A}$$

and

$${\psi }_1\left(\zeta \right)=\frac{1}{\alpha }{\lambda }^{\alpha /A}{q}_0{\left(1-p_0\right)}^{\alpha }\frac{1}{{\zeta }^{\alpha }},$$

where $A:=1+\frac{1}{\alpha }{q}_0{\lambda }^{\alpha }{\left(1-p_0\right)}^{\alpha }$.

From Theorem 1, Equation (20) is oscillatory if $\lambda <\mu$ and

$$q_0{\left(A\lambda \right)}^{\alpha }ln\frac{\mu }{\lambda }>{\left(1+\frac{p_0^{\beta }}{\mu }\right)}^{\beta /\alpha }\frac{1}{\kappa \mathrm{e}}.$$

(21)

Using Theorem 2, we have that if

$${\left(\frac{1}{\alpha }{\lambda }^{\alpha /A}{q}_0{\left(1-p_0\right)}^{\alpha }\right)}^{1/\alpha }>\frac{\alpha }{{\left(\alpha +1\right)}^{\left(\alpha +1\right)/\alpha }}$$

or

$$q_0{\lambda }^{\alpha /A}{\left(1-p_0\right)}^{\alpha }>\frac{{\alpha }^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}},$$

(22)

then (20) is oscillatory.

As a particular case, the known criteria for oscillation of equation

$${\left({\left({\left(x\left(\zeta \right)\right)}^{\prime }\right)}^{1/3}\right)}^{\prime }+\frac{q_0}{{\zeta }^{4/3}}{x}^{1/3}\left(\frac{9}{10}\zeta \right)=0,$$

(23)

where $q_0>0$, are:

Using Comparison Theory (As in [5])
C1	Corollary 2 in [5]	q0 < 3.61643
C2	Condition (21)	q0 < 1.92916
Using comparison theory (as in [11])
C3	Corollary 2.1 in [11]	q0 < 1.92916
By employing the Riccati transformation
C4	Corollary 2.1 in [16]	q0 < 0.16312
C5	Theorem 6 in [11]	q0 < 0.16243
C6	Condition (22)	q0 < 0.16131

Remark 1.

Note that, Theorem 1 improves Theorem 2 in [5]. The criterion of oscillation in Theorem 1 (as (21)) essentially takes into account the influence of delay argument $\tau \left(\zeta \right)$, which has been neglected in results of [11]. Moreover, Theorem 2 improves Theorem 6 in [11], and supports the most efficient condition for oscillation of Equation (20).

3. Oscillation Theorems in Non-Canonical Case

In the following, we derive new criteria for oscillation of solution of (1) in non-canonical case ${\theta }_{{\zeta }_0}\left(\infty \right)<\infty$. For convenience, we denote that: ${\theta }_{\zeta }\left(\infty \right):=\overline{\theta }\left(\zeta \right)$ and

$$\Theta \left(\zeta \right):=q\left(\zeta \right){\left(1-p\left(\sigma \left(\zeta \right)\right)\frac{\overline{\theta }\left(\tau \left(\sigma \left(\zeta \right)\right)\right)}{\overline{\theta }\left(\sigma \left(\zeta \right)\right)}\right)}^{\beta }.$$

Lemma 2.

Let $\Phi \left(\varrho \right)=L\varrho -M{\left(\varrho -N\right)}^{\left(\alpha +1\right)/\alpha },$ where $M>0,L$ and N are constants. Then, the maximum value of Φ on $\mathbb{R}$ is at ${\varrho }^{*}=N+{\left(\alpha L/\left(\left(\alpha +1\right)M\right)\right)}^{\alpha }$ and is given by

$$\underset{\varrho \in R}{max}\Phi \left(\varrho \right)=\Phi \left({\varrho }^{*}\right)=LN+\frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{L^{\alpha +1}}{M^{\alpha }}.$$

Lemma 3.

Let x be an eventually positive solution of (1), $z^{\prime }\left(\zeta \right)<0$ and

$${\int }_{{\zeta }_0}^{\infty }{\left(\frac{1}{r\left(\zeta \right)}{\int }_{{\zeta }_1}^{\zeta }\Theta \left(s\right)\mathrm{d}s\right)}^{1/\alpha }\mathrm{d}\zeta =\infty$$

(24)

holds. If there exists a constant $\delta \in \left[0,1\right)$ such that

$$\overline{\theta }\left(\zeta \right){\left(\eta \left(\zeta \right){\int }_{{\zeta }_0}^{\zeta }\Theta \left(s\right)\mathrm{d}s\right)}^{1/\alpha }\ge \delta ,$$

(25)

then

$$\frac{\mathrm{d}}{\mathrm{d}\zeta }\left(\frac{z\left(\zeta \right)}{{\overline{\theta }}^{\delta }\left(\zeta \right)}\right)\le 0.$$

(26)

Proof. 

Suppose that x positive solution of (1) and $z^{\prime }\left(\zeta \right)<0$. We assume that $x\left(\zeta \right)>0,$$x\left(\tau \left(\zeta \right)\right)>0$ and $x\left(\sigma \left(\zeta \right)\right)>0$ for $\zeta \ge {\zeta }_1,$ where ${\zeta }_1$ is sufficiently large. Then, from (1), we obtain $z\left(\zeta \right)>0$ and

$${\left(r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }\right)}^{\prime }=-q\left(\zeta \right)x^{\beta }\left(\sigma \left(\zeta \right)\right)<0.$$

(27)

Thus, we have

$$z\left(\zeta \right)\ge -{\int }_{\zeta }^{\infty }\frac{1}{r^{1/\alpha }\left(\varrho \right)}{r}^{1/\alpha }\left(\varrho \right)z^{\prime }\left(\varrho \right)\mathrm{d}\varrho \ge -\overline{\theta }\left(\zeta \right)r^{1/\alpha }\left(\zeta \right)z^{\prime }\left(\zeta \right)$$

(28)

and so

$$\frac{\mathrm{d}}{\mathrm{d}\zeta }\left(\frac{z\left(\zeta \right)}{\overline{\theta }\left(\zeta \right)}\right)=\frac{\overline{\theta }\left(\zeta \right)z^{\prime }\left(\zeta \right)+{\left(r\left(\zeta \right)\right)}^{-1/\alpha }z\left(\zeta \right)}{{\overline{\theta }}^2\left(\zeta \right)}\ge 0.$$

Hence, we find

$$x\left(\zeta \right)\ge \left(1-p\left(\zeta \right)\frac{\overline{\theta }\left(\tau \left(\zeta \right)\right)}{\overline{\theta }\left(\zeta \right)}\right)z\left(\zeta \right),$$

which with (27) gives

$${\left(r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }\right)}^{\prime }\le -\Theta \left(\zeta \right)z^{\beta }\left(\sigma \left(\zeta \right)\right).$$

(29)

Next, since z is a positive decreasing function, we have that ${lim}_{\zeta \to \infty }z\left(\zeta \right)=ϵ\ge 0$. Let $ϵ>0$. Then, from (29), there exists a ${\zeta }_1\ge {\zeta }_0$ such that

$${\left(r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }\right)}^{\prime }\le -\Theta \left(\zeta \right)z^{\beta }\left(\sigma \left(\zeta \right)\right)\le -{ϵ}^{\beta }\Theta \left(\zeta \right).$$

Integrating this inequality from ${\zeta }_1$ to $\zeta$, we have

$$-z^{\prime }\left(\zeta \right)\ge {ϵ}^{\beta /\alpha }{\left(\frac{1}{r\left(\zeta \right)}{\int }_{{\zeta }_1}^{\zeta }\Theta \left(s\right)\mathrm{d}s\right)}^{1/\alpha }.$$

(30)

Integrating (30) from ${\zeta }_1$ to $\zeta$, we get

$$z\left(\zeta \right)\le z\left({\zeta }_1\right)-{ϵ}^{\beta /\alpha }{\int }_{{\zeta }_1}^{\zeta }{\left(\frac{1}{r\left(\varrho \right)}{\int }_{{\zeta }_1}^{\varrho }\Theta \left(s\right)\mathrm{d}s\right)}^{1/\alpha }\mathrm{d}\varrho .$$

Taking ${lim}_{\zeta \to \infty }$ of this inequality and using (24), we arrive at a contradiction with positivity of z. Therefore, we get that $ϵ=0$ that is

$$\underset{\zeta \to \infty }{lim}x\left(\zeta \right)=\underset{\zeta \to \infty }{lim}z\left(\zeta \right)=0.$$

(31)

Integrating (29) from ${\zeta }_1$ to $\zeta$, we get

$$\begin{array}{ccc}r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }& \le & r\left({\zeta }_1\right){\left(z^{\prime }\left({\zeta }_1\right)\right)}^{\alpha }-{\int }_{{\zeta }_1}^{\zeta }\Theta \left(s\right)z^{\beta }\left(\sigma \left(s\right)\right)\mathrm{d}s\hfill \\ & \le & r\left({\zeta }_1\right){\left(z^{\prime }\left({\zeta }_1\right)\right)}^{\alpha }-z^{\beta }\left(\sigma \left(\zeta \right)\right){\int }_{{\zeta }_1}^{\zeta }\Theta \left(s\right)\mathrm{d}s.\hfill \end{array}$$

(32)

In view of (31), we see that

$$r\left({\zeta }_1\right){\left(z^{\prime }\left({\zeta }_1\right)\right)}^{\alpha }+z^{\beta }\left(\sigma \left(\zeta \right)\right){\int }_{{\zeta }_0}^{{\zeta }_1}\Theta \left(s\right)\mathrm{d}s>0,$$

(33)

for $\zeta \ge {\zeta }_2,$ where ${\zeta }_2$ large enough. Combining (32) and (33) and using the fact that $z^{\beta -\alpha }\left(\zeta \right)\ge \eta \left(\zeta \right)$, we find

$$r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }\le -z^{\beta }\left(\sigma \left(\zeta \right)\right){\int }_{{\zeta }_0}^{\zeta }\Theta \left(s\right)\mathrm{d}s\le -\eta \left(\zeta \right)z^{\alpha }\left(\zeta \right){\int }_{{\zeta }_0}^{\zeta }\Theta \left(s\right)\mathrm{d}s,$$

(34)

which with (25) gives

$$\begin{array}{ccc}{z}^{\prime }\left(\zeta \right)& \le & -r{\left(\zeta \right)}^{-1/\alpha }z\left(\zeta \right){\left(\eta \left(\zeta \right){\int }_{{\zeta }_0}^{\zeta }\Theta \left(s\right)\mathrm{d}s\right)}^{1/\alpha }\hfill \\ & \le & -\frac{\delta }{\overline{\theta }\left(\zeta \right)}r{\left(\zeta \right)}^{-1/\alpha }z\left(\zeta \right).\hfill \end{array}$$

Thus, we have

$$\frac{\mathrm{d}}{\mathrm{d}\zeta }\left(\frac{z\left(\zeta \right)}{{\overline{\theta }}^{\delta }\left(\zeta \right)}\right)=\frac{{\overline{\theta }}^{\delta -1}\left(\zeta \right)\left(\overline{\theta }\left(\zeta \right)z^{\prime }\left(\zeta \right)+\delta {\left(\mu \left(\zeta \right)r\left(\zeta \right)\right)}^{-1/\alpha }z\left(\zeta \right)\right)}{{\overline{\theta }}^{2\delta }\left(\zeta \right)}\le 0.$$

This completes the proof. □

Theorem 3.

Assume that (24) holds, $\overline{\theta }\left(\tau \left(\zeta \right)\right)\ge \overline{\theta }\left(\zeta \right)$ and there exists a $\delta \in \left[0,1\right)$ such that (25) holds. If there exist a function $\rho \in C^1\left(\left[{\zeta }_0,\infty \right),\left(0,\infty \right)\right)$ and a ${\zeta }_1\in \left[{\zeta }_0,\infty \right)$ such that

$$\underset{\zeta \to \infty }{limsup}\left\{\frac{{\overline{\theta }}^{\alpha }\left(\zeta \right)}{\rho \left(\zeta \right)}{\int }_{{\zeta }_1}^{\zeta }\left(\rho \left(\varrho \right)\Theta \left(\varrho \right)\eta \left(\varrho \right){\left(\frac{\overline{\theta }\left(\sigma \left(\varrho \right)\right)}{\overline{\theta }\left(\varrho \right)}\right)}^{\delta \beta }-\frac{r\left(\varrho \right){\left({\rho }^{\prime }\left(\varrho \right)\right)}^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}{\rho }^{\alpha }\left(\varrho \right)}\right)\mathrm{d}\varrho \right\}>1,$$

(35)

then all solutions of (1) are oscillatory.

Proof. 

Assuming that the required result is not fulfilled, we suppose, without loss of generality, that x is a positive solution of (1) on $\left[{\zeta }_0,\infty \right)$. Then, there exists ${\zeta }_1\ge {\zeta }_0$ such that $x\left(\tau \left(\zeta \right)\right)>0$ and $x\left(\sigma \left(\zeta \right)\right)>0$ for all $\zeta \ge {\zeta }_1.$ Obviously, for all $\zeta \ge {\zeta }_1,$$z\left(\zeta \right)\ge x\left(\zeta \right)>0$ and ${\left(r{\left(z^{\prime }\right)}^{\alpha }\right)}^{\prime }\le 0$. Therefore, $z^{\prime }\left(\zeta \right)$ is either eventually negative or eventually positive.

Assume first that $z^{\prime }\left(\zeta \right)<0.$ From Lemma 3, we get that (28) and (29) hold. We define the function

$$w:=\rho \left(\frac{r{\left(z^{\prime }\right)}^{\alpha }}{z^{\alpha }}+\frac{1}{{\overline{\theta }}^{\alpha }}\right).$$

(36)

Using (28), we see that $w\left(\zeta \right)\ge 0$ for all $\zeta \ge {\zeta }_2\ge {\zeta }_1$. Differentiating (36), we get

$$w^{\prime }=\frac{{\rho }^{\prime }}{\rho }w+\rho \frac{{\left(r{\left(z^{\prime }\right)}^{\alpha }\right)}^{\prime }}{z^{\alpha }}-\alpha \rho \frac{r{\left(z^{\prime }\right)}^{\alpha +1}}{z^{\alpha +1}}+\alpha \rho \frac{1}{r^{1/\alpha }{\overline{\theta }}^{\alpha +1}}.$$

From Lemma 1, (29) and (36), we obtain

$$w^{\prime }\le -\frac{\alpha }{{\left(\rho r\right)}^{1/\alpha }}{\left(w-\frac{\rho }{{\overline{\theta }}^{\alpha }}\right)}^{1+1/\alpha }-\rho \Theta \eta \frac{z^{\beta }\left(\sigma \right)}{z^{\beta }}+\alpha \rho \frac{1}{r^{1/\alpha }{\overline{\theta }}^{\alpha +1}}+\frac{{\rho }^{\prime }}{\rho }w.$$

(37)

Using Lemma 2 with $L:={\rho }^{\prime }/\rho ,$ $M:=\alpha /{\left(\rho r\right)}^{1/\alpha },$ $N=\rho /{\overline{\theta }}^{\alpha }$ and $\varrho :=w$, we get

$$\frac{{\rho }^{\prime }}{\rho }w-\frac{\alpha }{{\left(\rho r\right)}^{1/\alpha }}{\left(w-\frac{\rho }{{\overline{\theta }}^{\alpha }}\right)}^{1+1/\alpha }\le \frac{{\rho }^{\prime }}{{\overline{\theta }}^{\alpha }}+\frac{r}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{{\left({\rho }^{\prime }\right)}^{\alpha +1}}{{\rho }^{\alpha }},$$

which with (37) gives

$$w^{\prime }\le \frac{{\rho }^{\prime }}{{\overline{\theta }}^{\alpha }}+\frac{r}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{{\left({\rho }^{\prime }\right)}^{\alpha +1}}{{\rho }^{\alpha }}-\rho \Theta \eta \frac{z^{\beta }\left(\sigma \right)}{z^{\beta }}+\alpha \rho \frac{1}{r^{1/\alpha }{\overline{\theta }}^{\alpha +1}}.$$

(38)

From Lemma 3 we arrive at (26). From (26), we conclude that

$$\frac{z\left(\sigma \left(\zeta \right)\right)}{z\left(\zeta \right)}\ge \frac{{\overline{\theta }}^{\delta }\left(\sigma \left(\zeta \right)\right)}{{\overline{\theta }}^{\delta }\left(\zeta \right)}.$$

(39)

Thus, (38) becomes

$$w^{\prime }\le -\rho \Theta \eta {\left(\frac{\overline{\theta }\left(\sigma \left(\zeta \right)\right)}{\overline{\theta }\left(\zeta \right)}\right)}^{\delta \beta }+\frac{r}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{{\left({\rho }^{\prime }\right)}^{\alpha +1}}{{\rho }^{\alpha }}+{\left(\frac{\rho }{{\overline{\theta }}^{\alpha }}\right)}^{\prime }.$$

Integrating this inequality from ${\zeta }_2$ to $\zeta$, we have

$$\begin{array}{c}{\int }_{{\zeta }_2}^{\zeta }\left(\rho \left(\varrho \right)\Theta \left(\varrho \right)\eta \left(\varrho \right){\left(\frac{\overline{\theta }\left(\sigma \left(\varrho \right)\right)}{\overline{\theta }\left(\varrho \right)}\right)}^{\delta \beta }-\frac{r\left(\varrho \right){\left({\rho }^{\prime }\left(\varrho \right)\right)}^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}{\rho }^{\alpha }\left(\varrho \right)}\right)\mathrm{d}\varrho \hfill \\ \le {\left.\left(\frac{\rho \left(\zeta \right)}{{\overline{\theta }}^{\alpha }\left(\zeta \right)}-w\left(\zeta \right)\right)\right|}_{{\zeta }_2}^{\zeta }\hfill \\ \le {\left.-\left(\rho \left(\zeta \right)\frac{r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }}{z^{\alpha }\left(\zeta \right)}\right)\right|}_{{\zeta }_2}^{\zeta }.\hfill \end{array}$$

(40)

From (28), we have

$$-\frac{r{\left(\zeta \right)}^{1/\alpha }{z}^{\prime }\left(\zeta \right)}{z\left(\zeta \right)}\le \frac{1}{\overline{\theta }\left(\zeta \right)},$$

which in view of (40) implies

$$\frac{{\overline{\theta }}^{\alpha }\left(\zeta \right)}{\rho \left(\zeta \right)}{\int }_{{\zeta }_2}^{\zeta }\left(\rho \left(\varrho \right)\Theta \left(\varrho \right)\eta \left(\varrho \right){\left(\frac{\overline{\theta }\left(\sigma \left(\varrho \right)\right)}{\overline{\theta }\left(\varrho \right)}\right)}^{\delta \beta }-\frac{r\left(\varrho \right){\left({\rho }^{\prime }\left(\varrho \right)\right)}^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}{\rho }^{\alpha }\left(\varrho \right)}\right)\mathrm{d}\varrho \le 1.$$

Taking the lim sup on both sides of this inequality, we arrive at a contradiction with (35).

Now assume that $z^{\prime }\left(\zeta \right)>0.$ Since $z^{\prime }>0$ and $\overline{\theta }\left(\tau \left(\zeta \right)\right)\ge \overline{\theta }\left(\zeta \right).$ Then

$$x\left(\zeta \right)\ge \left(1-p\left(\zeta \right)\right)z\left(\zeta \right)\ge \left(1-p\left(\zeta \right)\frac{\overline{\theta }\left(\tau \left(\zeta \right)\right)}{\overline{\theta }\left(\zeta \right)}\right)z\left(\zeta \right).$$

(41)

From (1) and (41), we have

$$\begin{array}{ccc}{\left(r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }\right)}^{\prime }& \le & -q\left(\zeta \right){\left(1-p\left(\sigma \left(\zeta \right)\right)\frac{\overline{\theta }\left(\tau \left(\sigma \left(\zeta \right)\right)\right)}{\overline{\theta }\left(\sigma \left(\zeta \right)\right)}\right)}^{\beta }{z}^{\beta }\left(\sigma \left(\zeta \right)\right)\hfill \\ & \le & -\Theta \left(\zeta \right)z^{\beta }\left(\sigma \left(\zeta \right)\right).\hfill \end{array}$$

Integrating the above inequality from ${\zeta }_2$ to $\zeta$, we have

$$\begin{array}{ccc}r\left(\zeta \right){\left(z^{\prime }\left(\zeta \right)\right)}^{\alpha }& \le & r\left({\zeta }_2\right){\left(z^{\prime }\left({\zeta }_2\right)\right)}^{\alpha }-{\int }_{{\zeta }_2}^{\zeta }\Theta \left(s\right)z^{\beta }\left(\sigma \left(s\right)\right)\mathrm{d}s\hfill \\ & \le & r\left({\zeta }_2\right){\left(z^{\prime }\left({\zeta }_2\right)\right)}^{\alpha }-z^{\beta }\left(\sigma \left({\zeta }_2\right)\right){\int }_{{\zeta }_2}^{\zeta }\Theta \left(s\right)\mathrm{d}s.\hfill \end{array}$$

(42)

Now, from (24) and ${\theta }_{{\zeta }_0}\left(\infty \right)<\infty$, we get that ${\int }_{{\zeta }_1}^{\zeta }\Theta \left(s\right)\mathrm{d}s$ must be unbounded, that is

$${\int }_{{\zeta }_1}^{\infty }\Theta \left(s\right)\mathrm{d}s=\infty .$$

(43)

From (42) and (43), we get contradictions to $z^{\prime }>0$ as $\zeta \to \infty .$ The proof is complete. □

Corollary 1.

Assume that (24) holds, $\overline{\theta }\left(\tau \left(\zeta \right)\right)\ge \overline{\theta }\left(\zeta \right)$ and there exists a $\delta \in \left[0,1\right)$ such that (25) holds. If there exists a ${\zeta }_1\in \left[{\zeta }_0,\infty \right)$ such that

$$\underset{\zeta \to \infty }{limsup}{\int }_{{\zeta }_1}^{\zeta }\left({\overline{\theta }}^{\alpha }\left(\varrho \right)\Theta \left(\varrho \right)\eta \left(\varrho \right){\left(\frac{\overline{\theta }\left(\sigma \left(\varrho \right)\right)}{\overline{\theta }\left(\varrho \right)}\right)}^{\delta \beta }-{\left(\frac{\alpha }{\alpha +1}\right)}^{\alpha +1}\frac{1}{r^{1/\alpha }\left(\varrho \right)\overline{\theta }\left(\varrho \right)}\right)\mathrm{d}\varrho >1,$$

(44)

then all solutions of (1) are oscillatory.

Example 2.

Consider the second-order differential equation

$${\left({\zeta }^{\alpha +1}{\left({\left(x\left(\zeta \right)+p_{0}x\left(\kappa \zeta \right)\right)}^{\prime }\right)}^{\alpha }\right)}^{\prime }+q_0{x}^{\alpha }\left(\lambda \zeta \right)=0,\zeta \ge 1,$$

(45)

where $q_0>0$ and $\kappa ,\lambda \in \left(0,1\right]$. Note that, $r\left(\zeta \right)={\zeta }^{\alpha +1},p\left(\zeta \right)=p_0,$$\tau \left(\zeta \right)=\kappa \zeta ,q\left(\zeta \right)=q_0,\sigma \left(\zeta \right)=\lambda \zeta$ and $\beta =\alpha$. It is easy to conclude that $\eta \left(\zeta \right)=1,\overline{\theta }\left(\zeta \right)=\alpha /{\zeta }^{1/\alpha }$ and $\Theta \left(\zeta \right)=q_0{\left(1-2^{1/\alpha }{p}_0\right)}^{\alpha }$. We note that the conditions (24) and (25) hold. Furthermore, condition (44) becomes

$$\underset{\zeta \to \infty }{limsup}{\int }_{{\zeta }_0}^{\zeta }\left(\frac{{\alpha }^{\alpha }}{s}{q}_0{\left(1-2^{1/\alpha }{p}_0\right)}^{\alpha }\frac{1}{{\lambda }^{\delta }}-{\left(\frac{\alpha }{\alpha +1}\right)}^{\alpha +1}\frac{1}{\alpha s}\right)\mathrm{d}s>1.$$

Using Corollary 1, we get that every solution of Equation (45) is oscillatory if

$$q_0{\left(1-2^{1/\alpha }{p}_0\right)}^{\alpha }\frac{1}{{\lambda }^{\delta }}>\frac{1}{{\left(\alpha +1\right)}^{\alpha +1}}.$$

(46)

From Theorem 2.2 in [4], (45) is oscillatory if $\alpha \ge 1$ and

$$q_0{\left(1-2^{1/\alpha }{p}_0\right)}^{\alpha }>\frac{1-\left(1-\alpha \right){\left(\alpha +1\right)}^{\alpha +1}}{{\alpha }^{\alpha +1}{\left(\alpha +1\right)}^{\alpha +1}}.$$

(47)

By using Theorem 2.4 in [7], (45) is oscillatory if

$$q_0^{1/\alpha }\left(1-2^{1/\alpha }{p}_0\right)ln\frac{1}{\lambda }>\frac{1}{\mathrm{e}}.$$

(48)

Consider the special case in which $\alpha =1,$$\kappa =\lambda =1/2,$$p_0=1/4$ and $q_0=\sqrt{2}/3$, that is,

$${\left({\zeta }^2{\left({\left(x\left(\zeta \right)+\frac{1}{4}x\left(\frac{1}{2}\zeta \right)\right)}^{\prime }\right)}^{\alpha }\right)}^{\prime }+\frac{\sqrt{2}}{3}x\left(\frac{1}{2}\zeta \right)=0.$$

We note that the criteria arrive at $\sqrt{2}/3>0.4307,$$\sqrt{2}/3>0.5$ and $\sqrt{2}/3>1.0615$. Hence, it is obvious that (47) and (48) fail to apply.

4. Conclusions

During this work, we highlighted the oscillatory properties of solutions of differential Equation (1). By using many techniques, we have created new criteria that are more effective than the relevant criteria in the literature. Moreover, we discussed oscillatory behavior in both canonical and non-canonical cases. Through the examples, it turns out that our results improve and complete some of the results in [4,5,7,11,16]. Finally, we can try to extend our results to differential equations with a damping term, in the future.